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Flats' body is shown to be somewhat two-dimensional, hence his name.
He first appears as a cameo in the episode "Sandy's Rocket," where he was one of SpongeBob's "alien hunting" victims. He also makes another minor appearance in "Texas," where he attends Sandy's Texas party.
His only major appearance, however, is in the episode "The Bully." He joins Mrs. Puff's Boating School and wants to "kick SpongeBob's butt." The latter takes this extremely seriously, reacting as if he were going to actually kill him. He makes many attempts to reason with and/or escape him, but to no avail. He even calls Patrick to protect him, only to find out that Flats is friends with him and that they were community college buddies.
He also tries to go to Mrs. Puff to help, who talks to Flats (much to SpongeBob's dismay) and explains to SpongeBob that "kicking his butt" means that Flats wants to become his friend, which is obviously not true. He then encounters Flats' Dad in a boat and confronts him about his son's actions, only for the latter to go up to his father and teach him a lesson about talking to strangers, threatening to harm one of his parents.
Flats goes so far as to chase SpongeBob out of school into town, even trying to run him over with a garbage truck. However, said garbage truck slips on a banana peel and flips over, critically injuring Flats in the process. He wakes up in the hospital and learns that SpongeBob saved his life through CPR, yet he still wants to kick his butt. Eventually, Flats corners SpongeBob in his own home, and proceeds to "kick his butt" by repeatedly hitting him. However, SpongeBob discovers that his spongy body simply absorbs the blows and does not harm him in any way at all.
Flats continues to hit SpongeBob throughout the whole week until he finally passes out from exhaustion one morning at boating school.
Flats is a flounder, which is a species of flatfish, who have both eyes on one half of their body.